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2. On unitary algebras with graded involution of quadratic growth.
- Author
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Bessades, D.C.L., Costa, W.D.S., and Santos, M.L.O.
- Subjects
- *
ALGEBRA , *SUPERALGEBRAS , *POLYNOMIALS - Abstract
Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Drazin and group invertibility in algebras spanned by two idempotents.
- Author
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Biswas, Rounak and Roy, Falguni
- Subjects
- *
GROUP algebras , *IDEMPOTENTS , *ASSOCIATIVE algebras , *COMPLEX numbers , *ALGEBRA , *REAL numbers , *ASSOCIATIVE rings - Abstract
For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Graded identities with involution for the algebra of upper triangular matrices.
- Author
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Diniz, Diogo, Ramos, Alex, and Galdino, José Lucas
- Subjects
- *
ALGEBRA , *MATRICES (Mathematics) - Abstract
Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Some remarks on a paper by L. Carlitz
- Author
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Dominici, Diego
- Subjects
- *
POLYNOMIALS , *ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
6. Triangular norms. Position paper I: basic analytical and algebraic properties
- Author
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Klement, Erich Peter, Mesiar, Radko, and Pap, Endre
- Subjects
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NILPOTENT groups , *TRIANGLES , *SET theory , *ALGEBRA - Abstract
We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
7. A characterization of the natural grading of the Grassmann algebra and its non-homogeneous [formula omitted]-gradings.
- Author
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Fideles, Claudemir, Gomes, Ana Beatriz, Grishkov, Alexandre, and Guimarães, Alan
- Subjects
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ALGEBRA , *POLYNOMIALS , *LOGICAL prediction , *SUPERALGEBRAS , *C*-algebras - Abstract
Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
8. Maps preserving matrices of local reduced minimum modulus zero at a fixed vector.
- Author
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Bourhim, Abdellatif, Mabrouk, Mohamed, and Mbekhta, Mostafa
- Subjects
- *
MATRICES (Mathematics) , *LINEAR operators , *ALGEBRA - Abstract
Let n be an integer greater than 1, and M n (C) be the algebra of all n × n -complex matrices. Let x 0 ∈ C n be a nonzero vector, and Φ be a linear map on M n (C) such that Φ (I) is invertible. For any matrix T ∈ M n (C) , let γ (T , x 0) denote the local reduced minimum modulus of T at x 0. In this paper, we show that Φ satisfies γ (T , x 0) = 0 ⇔ γ (Φ (T) , x 0) = 0 , (T ∈ M n (C)) , if and only if there are two invertible matrices A , B ∈ M n (C) such that A x 0 = A ⁎ x 0 = x 0 and Φ (T) = B T A for all T ∈ M n (C). When n = 2 , we show that the invertibility hypothesis of Φ (I) is redundant. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Regularity of interval max-plus matrices.
- Author
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Myšková, Helena and Plavka, Ján
- Subjects
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MATRICES (Mathematics) , *LINEAR systems , *ALGEBRA - Abstract
Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A ⊗ x = b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
10. Some sixth-order variants of Ostrowski root-finding methods
- Author
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Chun, Changbum and Ham, YoonMee
- Subjects
- *
PAPER , *EQUATIONS , *ALGEBRA , *MATHEMATICS - Abstract
Abstract: In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
11. AN UNBOUNDED GENERALIZATION OF TOMITA's OBSERVABLE ALGEBRAS II.
- Author
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Inoue, Hiroshi
- Subjects
- *
OPERATOR algebras , *ALGEBRA , *GENERALIZATION - Abstract
In a previous paper [4] we tried to build the basic theory of unbounded Tomita's observable algebras called T †-algebras which are related to unbounded operator algebras, especially unbounded Tomita-Takesaki theory, operator algebras on Krein spaces, studies of positive linear functionals on *-algebras and so on. And we defined the notions of regularity, semisimplicity and singularity of T †-algebras and characterized them. In this paper we shall proceed further with our studies of T †-algebras and investigate whether a T †-algebra is decomposable into a regular part and a singular part. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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12. A note on the paper in CAGD (2004, 21 (2), 181–191)
- Author
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Zhang, Ren-Jiang and Wang, Guo-Jin
- Subjects
- *
POLYNOMIALS , *APPROXIMATION theory , *FUNCTIONAL analysis , *ALGEBRA - Abstract
Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
13. Polynomial identities and images of polynomials on null-filiform Leibniz algebras.
- Author
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de Mello, Thiago Castilho and Souza, Manuela da Silva
- Subjects
- *
POLYNOMIALS , *ALGEBRA , *MULTILINEAR algebra , *VECTOR spaces , *C*-algebras - Abstract
In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If L n is an n -dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id (L n) , the polynomial identities of L n , and we explicitly compute the images of multihomogeneous polynomials on L n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of L n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for L n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
14. On Construction of Darboux integrable discrete models.
- Author
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Zheltukhin, Kostyantyn and Zheltukhina, Natalya
- Subjects
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INTEGRAL equations , *ALGEBRA - Abstract
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
15. Quasi-Whittaker modules for the n-th Schrödinger algebra.
- Author
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Chen, Zhengxin and Wang, Yu
- Subjects
- *
UNIVERSAL algebra , *ALGEBRA , *LIE algebras , *C*-algebras - Abstract
The n -th Schrödinger algebra sch n defined in [14] is the semi-direct product of the Lie algebra sl 2 with the n -th Heisenberg Lie algebra h n , which generalizes the Schrödinger algebra sl 2 ⋉ h 1. Let ϕ : h n → C be a nonzero Lie algebra homomorphism. A sch n -module V is called quasi-Whittaker of type ϕ if V = U (sch n) v , where U (sch n) is the universal enveloping algebra of sch n , v is a nonzero vector such that x v = ϕ (x) v for any x ∈ h n. In this paper, we prove that a simple sch n -module V is a quasi-Whittaker module if and only if V is a locally finite h n -module. Then we classify the simple quasi-Whittaker modules of ϕ , according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
16. Differential codimensions and exponential growth.
- Author
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Rizzo, Carla
- Subjects
- *
ASSOCIATIVE algebras , *LIE algebras , *DIFFERENTIAL algebra , *ALGEBRA , *POLYNOMIALS , *VARIETIES (Universal algebra) , *EXPONENTIAL sums - Abstract
Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L (A) = lim n → ∞ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L (A) coincides with exp (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
17. Universal enveloping of a graded Lie algebra.
- Author
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Yasumura, Felipe Yukihide
- Subjects
- *
UNIVERSAL algebra , *ABELIAN groups , *ALGEBRA - Abstract
In this paper we construct a graded universal enveloping algebra of a G -graded Lie algebra, where G is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
18. Order isomorphisms on effect algebras of the C⁎-algebras of type [formula omitted].
- Author
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Abdelali, Zine El Abidine and El Khatiri, Youssef
- Subjects
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ISOMORPHISM (Mathematics) , *ALGEBRA , *HILBERT space , *HAUSDORFF spaces , *COMMERCIAL space ventures , *COMPACT spaces (Topology) , *C*-algebras , *LINEAR operators - Abstract
Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
19. An unbounded generalization of tomita's observable algebras III.
- Author
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Inoue, Hiroshi
- Subjects
- *
GENERALIZATION , *ALGEBRA - Abstract
In this paper we shall continue the studies of T †-algebras done in [ 8, 9 ], and above all we investigate decompositions of the vector representation part of a T †-algebra and apply the results to invariant positive sesquilinear forms on *-algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
20. A decidable theory involving addition of differentiable real functions.
- Author
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Buriola, Gabriele, Cantone, Domenico, Cincotti, Gianluca, Omodeo, Eugenio G., and Spartà, Gaetano T.
- Subjects
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DIFFERENTIABLE functions , *MATHEMATICAL analysis , *REAL numbers , *DERIVATIVES (Mathematics) , *ALGEBRA , *REAL variables - Abstract
This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
21. Comments on the paper "On the κ-Dirac oscillator revisited".
- Author
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Chargui, Yassine
- Subjects
- *
ALGEBRA , *EIGENVALUES , *EQUATIONS - Abstract
In Ref. [1] , the κ -Dirac equation, based on the κ -deformed Poincaré-Hopf algebra, have been studied. In particular, solutions of the κ -Dirac oscillator (DO), in a three-dimensional space, were obtained by deriving the associated radial equations. We point out, however, a miscalculation in treating these equations, which had led to erroneous conclusions, particularly, about the energy eigenvalues and the breaking of their infinite degeneracy by the κ -deformation. By the way, we present a simple alternative method to solve the problem using an algebraic procedure. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
22. Identities for subspaces of a parametric Weyl algebra.
- Author
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Lopatin, Artem and Rodriguez Palma, Carlos Arturo
- Subjects
- *
ALGEBRA , *POLYNOMIALS , *FINITE fields - Abstract
In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. Ring derivations of Murray–von Neumann algebras.
- Author
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Huang, Jinghao, Kudaybergenov, Karimbergen, and Sukochev, Fedor
- Subjects
- *
VON Neumann algebras , *ALGEBRA - Abstract
Let M be a type II 1 von Neumann algebra, S (M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S (M) with M ⊆ A. We prove that any ring derivation D from A into S (M) is necessarily inner. Further, we prove that if A is an E W ⁎ -algebra such that its bounded part A b is a W ⁎ -algebra without finite type I direct summands, then any ring derivation D from A into L S (A b) — the algebra of all locally measurable operators affiliated with A b , is an inner derivation. We also give an example showing that the condition M ⊆ A is essential. At the end of this paper, we provide several criteria for an abelian extended W ⁎ -algebra such that all ring derivations on it are linear. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. Maps preserving the ascent and descent of product of operators.
- Author
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Hosseinzadeh, Roja and Petek, Tatjana
- Subjects
- *
BANACH spaces , *COMMERCIAL space ventures , *ALGEBRA - Abstract
Let B (X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim X ≥ 3. In this paper, we characterize the maps from B (X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
25. Exponential entropy on sequential effect algebras.
- Author
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Singh, Akhilesh Kumar
- Subjects
- *
ENTROPY , *ALGEBRA , *ISOMORPHISM (Mathematics) , *DYNAMICAL systems - Abstract
The present paper introduces a new definition of entropy on sequential effect algebras. Unlike the logarithmic behaviour of the entropy defined in the literature, the entropy considered here is of exponential nature. Conditional entropy and entropy on the dynamical system are also introduced and studied. It is also proved that entropy of the dynamical system is invariant under isomorphism. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Preservation of properties of residuated algebraic structure by structures for the partial fuzzy set theory.
- Author
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Cao, Nhung and Štěpnička, Martin
- Subjects
- *
SET theory , *FUZZY sets , *RESIDUATED lattices , *FUZZY logic , *ALGEBRA , *AXIOMS - Abstract
This paper addresses the preservation of numerous essential properties of a residuated lattice structure in extended algebras for partial fuzzy set theory and partial fuzzy logics. The preservation includes the residuated lattice axioms, the identities narrowing the classes of the residuated lattices, and some well-known additional properties. In this paper, we consider nine algebras for partial fuzzy logics which incorporate handling undefined values in a bit different way. In particular, we consider the Bochvar, the Bochvar external, the Sobociński, the Kleene, the McCarthy, the Nelson, and the Łukasiewicz algebras, and two recently developed ones, namely the Lower estimation and the Dragonfly algebras. We summarize the obtained results in a comprehensible form which allows readers to easily check the information for the preserved and non-preserved properties in a certain partial algebraic structure. The resulting shape of the contribution is a sort of "atlas book" that aims at providing researchers with a comfortable and comprehensible form of an overview of the (non)preservation of fundamental properties of residuated structures. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
27. Graded identities of Mn(E) and their generalizations over infinite fields.
- Author
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Fidelis, Claudemir
- Subjects
- *
MATRICES (Mathematics) , *INFINITE groups , *ALGEBRA , *GENERALIZATION , *POLYNOMIALS , *COMMUTATIVE algebra , *TENSOR products - Abstract
Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
28. Matrix theory for independence algebras.
- Author
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Araújo, João, Bentz, Wolfram, Cameron, Peter J., Kinyon, Michael, and Konieczny, Janusz
- Subjects
- *
UNIVERSAL algebra , *ALGEBRA , *ENDOMORPHISMS , *VECTOR spaces , *SET theory , *MODEL theory - Abstract
A universal algebra A with underlying set A is said to be a matroid algebra if (A , 〈 ⋅ 〉) , where 〈 ⋅ 〉 denotes the operator subalgebra generated by , is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n , with at least two elements. Denote by End (A) the monoid of endomorphisms of A. In the 1970s, Głazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabó in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabó to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
29. Frame-valued Scott open set monad and its algebras.
- Author
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Liu, Mengying, Yue, Yueli, and Wei, Xiaowei
- Subjects
- *
ALGEBRA , *MONOIDS , *MONADS (Mathematics) - Abstract
For a frame Q , this paper studies the Kleisli monoids and Eilenberg-Moore algebras with respect to the Q -Scott open set monad. Firstly, we show that the Q -Scott open set functor can form a monad, and prove that the Kleisli monoids of this monad are exactly the algebraic Q -ordered closure spaces or equivalently the strong Q -convex spaces. Then we show that the Eilenberg-Moore algebras of Q -Scott open set monad are precisely the algebraic Q -modules or equivalently the fuzzy frames. Finally, we construct the strong Q -Scott open set monad, and study the corresponding Kleisli monoids and the Eilenberg-Moore algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. On polynomials satisfying power inequality for numerical radius.
- Author
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Dadar, Elham and Alizadeh, Rahim
- Subjects
- *
POLYNOMIALS , *ALGEBRA , *C*-algebras , *POLYNOMIAL rings - Abstract
Let A be a unital C ⁎ algebra and for every a ∈ A , r (a) denote the numerical radius of a ∈ A. The power inequality for numerical radius states that for every polynomial P (z) = z n and a ∈ A the inequality P (r (a)) ≥ r (P (a)) holds. In this paper, we get a characterization of polynomials with real coefficients that satisfy the power inequality on all 2 × 2 matrices with real entries. We also characterize all polynomials that satisfy the power inequality on every commutative unital C ⁎ algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. Cellular resolutions of monomial ideals and their Artinian reductions.
- Author
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Faridi, Sara, Farrokhi D.G., Mohammad D.G., Ghorbani, Roya, and Yazdan Pour, Ali Akbar
- Subjects
- *
BETTI numbers , *MORSE theory , *ALGEBRA - Abstract
The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. The variety of modal weak Gödel algebras.
- Author
-
Celani, Sergio A., Nagy, Agustín L., and San Martín, Hernán J.
- Subjects
- *
HEYTING algebras , *ALGEBRA , *CONGRUENCE lattices - Abstract
An algebra 〈 A , ∧ , ∨ , → , □ , 0 , 1 〉 of type (2 , 2 , 2 , 1 , 0 , 0) is said to be a modal weak Heyting algebra if 〈 A , ∧ , ∨ , → , 0 , 1 〉 is a weak Heyting algebra and the following conditions are satisfied for every a , b ∈ A : M1) □ (1) = 1 , M2) □ (a ∧ b) = □ (a) ∧ □ (b) and M3) □ (a → b) ≤ □ (a) → □ (b). If this algebra satisfies the inequality a ∧ (a → b) ≤ b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. On the subalgebra lattice of a restricted Lie algebra.
- Author
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Páez-Guillán, Pilar, Siciliano, Salvatore, and Towers, David A.
- Subjects
- *
ALGEBRA - Abstract
In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. A logical characterization of multi-adjoint algebras.
- Author
-
Cornejo, M. Eugenia, Fariñas del Cerro, Luis, and Medina, Jesús
- Subjects
- *
MATHEMATICAL logic , *ALGEBRA , *PROPOSITION (Logic) , *COMPLETENESS theorem , *FUZZY logic , *LOGIC , *AXIOMS - Abstract
This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hájek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
35. Rough set models of some abstract algebras close to pre-rough algebra.
- Author
-
Sardar, Masiur Rahaman and Chakraborty, Mihir Kumar
- Subjects
- *
ABSTRACT algebra , *ROUGH sets , *ALGEBRA , *MATHEMATICAL logic - Abstract
Rough set theory has already been algebraically investigated for decades and quasi-Boolean algebra has formed a basis for a number of structures emerging out of rough sets. Pre-rough algebra is one such algebra amongst them. A number of structures based on quasi-Boolean algebra but weaker than pre-rough algebra already exist. In this paper some algebras and their logics are added. Rough set models of the newly created algebras and some of the existing algebras are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. The classification of nilpotent Lie-Yamaguti algebras.
- Author
-
Abdelwahab, Hani, Barreiro, Elisabete, Calderón, Antonio J., and Fernández Ouaridi, Amir
- Subjects
- *
LIE algebras , *ALGEBRA , *CLASSIFICATION - Abstract
In this paper, we consider a generalization of the classical Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Lie-Yamaguti algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Lie-Yamaguti algebras, and we obtain from it the algebraic classification of the nilpotent Lie-Yamaguti algebras up to dimension four. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
37. On the normalizer of the reflexive cover of a unital algebra of linear transformations.
- Author
-
Bračič, Janko and Kandić, Marko
- Subjects
- *
JORDAN algebras , *BIVECTORS , *ALGEBRA - Abstract
Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V , in this paper we study the set Col (A) consisting of those invertible linear transformations S on V which map every subspace M ∈ Lat (A) to a subspace S M ∈ Lat (A). We show that Col (A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A , we give the complete description of Col (A). By using the primary decomposition of A , we first reduce the problem of characterizing Col (A) to the problem of characterizing the group Col (N) of a given nilpotent linear transformation N. While Col (N) always contains all invertible linear transformations of the commutant (N) ′ of N , it is always contained in the reflexive cover of (N) ′. We prove that Col (N) is a proper subgroup of (Alg Lat (N) ′) − 1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col (J 2 ⊕ J 2). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. Corrigendum to "The Green polynomials via vertex operators" [J. Pure Appl. Algebra 226 (2022) 107032].
- Author
-
Jing, Naihuan and Liu, Ning
- Subjects
- *
POLYNOMIALS , *ALGEBRA - Abstract
The statement of Proposition 3.1 in the cited paper misses a condition that the initial index i j must be chosen with care. Subsequent Theorem 3.2 is also modified accordingly. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Volumetric nonwoven structures: An algebraic framework for systematic design of infinite polyhedral frames using nonwoven fabric patterns.
- Author
-
Yildiz, Tolga and Akleman, Ergun
- Subjects
- *
NONWOVEN textiles , *FLEXIBLE structures , *STRUCTURAL frames , *ALGEBRA , *POLYHEDRA - Abstract
In this paper, we present an algebraic framework that can be used to construct a large class of 3D shapes and structures that can potentially provide unusual material properties. We formalized this framework as a 3D generalization of planar nonwoven textile structures that are used to mimic the woven structures. Our extension is based on the fact that it is straightforward to extend planar nonwoven textile structures into volumetric nonwoven textile structures, which we also call nonwoven volumetric fabrics. This property is essential because such an extension is impossible with planar woven structures. In other words, using this approach, it can be possible to easily produce volumetric structures that mimic the fabric behavior as if they were planar nonwoven textile structures, which is impossible to produce. These volumetric structures also correspond to regular & semiregular frame structures and are capable of representing previously unknown infinite regular polyhedra and flexible wood structures. [Display omitted] • We present a shape algebra using an extension of p-adic numbers to represent and design volumetric nonwoven structures. • This algebra leads to simple and powerful and topologically robust algorithms for modeling volumetric nonwoven structures. • We developed a tensor-based binary numbering scheme to uniquely describe connections. • We develop an algebraic framework by introducing 2 K -adic and 8 K -adic zeros (binary and octal). • We identified algebraic properties of 2 K -adic and 8 K -adic zeros and demonstrated how to use these properties to produce symmetric nonwoven structures. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Deformations, cohomologies and abelian extensions of compatible 3-Lie algebras.
- Author
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Hou, Shuai, Sheng, Yunhe, and Zhou, Yanqiu
- Subjects
- *
ALGEBRA , *LIE algebras , *COHOMOLOGY theory - Abstract
In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Reflexive modules over the endomorphism algebras of reflexive trace ideals.
- Author
-
Endo, Naoki and Goto, Shiro
- Subjects
- *
LOCAL rings (Algebra) , *INDECOMPOSABLE modules , *ALGEBRA , *ENDOMORPHISM rings , *COHEN-Macaulay rings , *ISOMORPHISM (Mathematics) , *ENDOMORPHISMS - Abstract
In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. A distant descendant of the six-vertex model.
- Author
-
Bazhanov, Vladimir V. and Sergeev, Sergey M.
- Subjects
- *
YANG-Baxter equation , *PARTITION functions , *SPIN-spin interactions , *STATISTICAL mechanics , *TRIANGLES , *STATISTICAL models , *ALGEBRA - Abstract
In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local "spin variables" can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent "dual" formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra U q (s l ˆ (2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight.
- Author
-
Song, Chao, Wang, Kai, and Zhang, Yuanyuan
- Subjects
- *
ALGEBRA , *COHOMOLOGY theory - Abstract
In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ , which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L ∞ [ 1 ] -algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ , the corresponding twisted L ∞ [ 1 ] -algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Modified bosonic integrable hierarchy.
- Author
-
Zhang, Yuanyuan, Cheng, Jipeng, Shen, Shoufeng, and Hu, Juan
- Subjects
- *
ALGEBRA , *EQUATIONS , *BOSONS - Abstract
In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l − l ′) –th modified KP hierarchy. Firstly, the representation of g l ˆ ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Rough L-fuzzy sets: Their representation and related structures.
- Author
-
Gégény, Dávid and Radeleczki, Sándor
- Subjects
- *
ROUGH sets , *FUZZY sets , *ALGEBRA - Abstract
The combination of fuzzy set theory and rough set theory has been discussed in a lot of research papers over the years. In this paper, we examine one such combination, namely the notion of rough L -fuzzy sets. We provide a representation theorem that determines when a pair of L -fuzzy sets is a rough L -fuzzy set, and we establish a connection between the lattice of rough fuzzy sets and the lattice of rough relations. Furthermore, we investigate the properties of the lattice of rough L -fuzzy sets and characterize the case when a three-valued Łukasiewicz-algebra can be defined on it. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
46. Generalized eigenproblem of interval max-min (fuzzy) matrices.
- Author
-
Plavka, Ján and Gazda, Matej
- Subjects
- *
MATRICES (Mathematics) , *MAXIMA & minima , *ALGEBRA - Abstract
In this paper we consider the generalized eigenproblem in max-min (fuzzy) algebra, i.e. given matrices A , B find a vector x and a constant λ such that A x = λ B x where the standard pair of operations, plus and times, have been replaced by the operations maximum and minimum. The entries of the vector or matrix are, in practice, usually not exact numbers and can rather be considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the strong, the universal, the L-controllable and the R-controllable generalized eigenproblems in max-min (fuzzy) algebra are presented. As a consequence of the obtained results, efficient algorithms for checking all equivalent conditions are introduced. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
47. Reductions of a covering approximation space from algebraic points of view.
- Author
-
Zhao, Zhengang
- Subjects
- *
ALGEBRAIC spaces , *MATRIX multiplications , *ALGEBRA , *ROUGH sets - Abstract
We first make a great improvement on the early definition of a reducible element of a covering approximation space and formulate the definition of a reduction of the covering. Then the crucial links between the two concepts are established and a method for obtaining an optimal reduction of the covering is proposed. More importantly, we introduce the representation matrix for a finite covering approximation space and define a new type of operation on matrices. To obtain a reduction of the covering, we need only to deal with the representation matrix, so that the reduction of the covering can be executed on computers. The concept of the product of two covering approximation spaces is defined in this paper and we explore the connections between the reductions of the covering of the product space and those of the coverings of the factor spaces. Furthermore, we deal with the connections from the standpoint of algebra. A new type of matrix product is defined and the property of the product is explored. Drawing on the product, we can research the reduction of the product space. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. A relational database model and algebra integrating fuzzy attributes and probabilistic tuples.
- Author
-
Cao, T.H.
- Subjects
- *
SET theory , *PROBABILITY theory , *PROBABILISTIC databases , *ALGEBRA , *FUZZY sets , *RELATIONAL databases , *RELATION algebras , *MULTISENSOR data fusion - Abstract
Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
49. Almost strongly fuzzy bounded operators with applications to fuzzy spectral theory.
- Author
-
Bînzar, Tudor
- Subjects
- *
SPECTRAL theory , *VECTOR spaces , *ALGEBRA , *NORMED rings - Abstract
In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Regularity and normality of (L,M)-fuzzy topological spaces using residual implication.
- Author
-
Liang, Chengyu
- Subjects
- *
METRIC spaces , *DISTRIBUTIVE lattices , *TOPOLOGICAL spaces , *AXIOMS , *ALGEBRA - Abstract
In this paper, the notions of regularity and normality of (L , M) -fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that (L , M) -fuzzy interior operator and (L , M) -fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an (L , M) -fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an (L , M) -fuzzy metric space. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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